Samuel J. Wheeler defends Wittgenstein's criticism of Cantor's set theory against the objections raised by Hilary Putnam. Putnam claims that Wittgenstein's dismissal of the basic tenets of this set theory concerning the noncountability of the set of real numbers was unfounded and ill‐conceived. In Wheeler's view, Putnam's charges result from his failure to grasp Wittgenstein's intention and, in particular, to consider the difference between empirical and logical impossibility. In my paper, I argue that Wheeler's defence is unsuccessful and, at the (...) same time, that Putnam's objections go too far. (shrink)

The paper discusses a relatively underexamined element of Wittgenstein’s philosophy of mathematics associated with his critique of set theory. I outline Wittgenstein’s objections to the theories of Dedekind and Cantor, including the confounding of extension and intension, the faulty definition of the infinite set as infinite extension and the critique of Cantor’s diagonal proof. One of Wittgenstein’s major objections to set theory was that the concept of the size of infinite sets, which Cantor expressed by means of symbols אₒ and (...) c, had no application, i.e. that there was no grammatical technique that could show how such expressions were to be used. Notions of set theory are, so to speak, exterior – they find themselves outdoors, outside of what we usually do. They form a discourse that takes us beyond the horizon of everydayness and commonality. They are like an engine idling of the language of mathematics. (shrink)

Maria Kokoszyńska-Lutmanowa (1906-1981) was a Polish philosopher and student of Kazimierz Twardowski, the founder of the Lvov-Warsaw School. In 1938 she went to the University of Cambridge (Newnham College) on a Sarah Smitton Fellowship. There she attended George Edward Moore’s lectures as well as one of Ludwig Wittgenstein’s Lectures on the Foundations of Mathematics in 1939. In this interview which was conducted with Alois Eder she talks about her encounter with Wittgenstein. It was published in 1978 in the Polish art (...) and culture magazine Odra and is her only published memoir about Wittgenstein. The interview is republished here both in English translation and as facsimile of the original publication with the kind permission of the Odra magazine. (shrink)

The aim of the article is to present Marek Siemek’s interpretation of modernity, focusing on problems related to understanding of the modern subject that arose from the reading of Max Horkheimer and Theodor W. Adorno’s Dialectic of Enlightenment. Siemek seems to endorse a general drive of Habermas’ theory of intersubjective communication intended to overcome the dialectics of Enlightenment and to complete the project of modernity. However, his position is that its foundation can be traced back to the philosophies of Fichte (...) and Hegel and their mutually complementary intersubjectivity models. Siemek seeks to reconcile the idea of the philosophy of intersubjectivity underlying Fichte’s and Hegel’s philosophies with the tenets of the philosophy of consciousness. (shrink)

This article aims to analyse Wittgenstein’s 1929–1932 notes concerning Frege’s critique of what is referred to as old formalism in the philosophy of mathematics. Wittgenstein disagreed with Frege’s critique and, in his notes, outlined his own assessment of formalism. First of all, he approvingly foregrounded its mathematics-game comparison and insistence that rules precede the meanings of expressions. In this article, I recount Frege’s critique of formalism and address Wittgenstein’s assessment of it to show that his remarks are not so much (...) a critique of Frege as rather a defence of the formalist anti-metaphysical investment. (shrink)

Spanning the period between Wittgenstein's return to Cambridge in 1929 and the first version of Philosophical Investigations in 1936, Piotr Dehnel explores the middle stage in Ludwig Wittgenstein's philosophical development and identifies the major issues which engrossed him, including phenomenology, philosophy of mathematics and philosophy of language. Contrary to the dominant perspective, Dehnel argues that this period was intrinsically different from the early and late stages and should not be viewed as a mere transitional phase. The distinctiveness (...) of Wittgenstein's middle work can be seen in his philosophical thinking as it unfolds in a non-linear trajectory: thoughts do not follow upon each other, ideas do not appear sequentially one by one, and insights do not form a straight chain. Dehnel portrays the diffused and multifarious quality of Wittgenstein's middle thinking, enabling readers to form a more comprehensive view of his entire philosophy and acquire a better grasp of his conceptual trajectory, complete with the intricacies and challenges that it poses. (shrink)

This is a reply to ‘Defending Wittgenstein’, Piotr Dehnel's critique of my article, ‘Defending Wittgenstein's Remarks on Cantor from Putnam’. I first show that my position is much more in agreement with Felix Mühlhölzer than Dehnel takes it to be, and that his criticism of me is nothing more than a failure to recognize this. I then show how Dehnel incorrectly reads Wittgenstein as rejecting set theory as false. It is an overemphasis on and a much (...) too narrow picture of ‘applicability’ which leads him to this view. Finally, I conclude by rejecting Dehnel's view that Wittgenstein was a finitist about mathematics. (shrink)

Topic modeling—a text‐mining technique often used to uncover thematic structures in large collections of texts—has been increasingly frequently used in the context of the analysis of scholarly output. In this study, we construct a corpus of 19,488 texts published since 1971 in seven leading journals in the field of bioethics and philosophy of medicine, and we use a machine learning algorithm to identify almost 100 topics representing distinct themes of interest in the field. On the basis of intertopic correlations, we (...) group the content‐based topics into eight clusters, thus providing a novel, fine‐grained intellectual map of bioethics and philosophy of medicine. Moreover, we conduct a number of diachronic analyses, examining how the “prominence” of different topics has changed across time. In this way, we are able to observe the distinct patterns in which bioethics and philosophy of medicine have evolved and changed their focus over the past half a century. (shrink)

Predictive processing (PP) has been repeatedly presented as a unificatory account of perception, action, and cognition. In this paper, we argue that this is premature: As a unifying theory, PP fails to deliver general, simple, homogeneous, and systematic explanations. By examining its current trajectory of development, we conclude that PP remains only loosely connected both to its computational framework and to its hypothetical biological underpinnings, which makes its fundamentals unclear. Instead of offering explanations that refer to the same set of (...) principles, we observe systematic equivocations in PP‐based models, or outright contradictions with its avowed principles. To make matters worse, PP‐based models are seldom empirically validated, and they are frequently offered as mere just‐so stories. The large number of PP‐based models is thus not evidence of theoretical progress in unifying perception, action, and cognition. On the contrary, we maintain that the gap between theory and its biological and computational bases contributes to the arrested development of PP as a unificatory theory. Thus, we urge the defenders of PP to focus on its critical problems instead of offering mere re‐descriptions of known phenomena, and to validate their models against possible alternative explanations that stem from different theoretical assumptions. Otherwise, PP will ultimately fail as a unified theory of cognition. (shrink)

Citizen identification systems (known also as ‘ID card systems’, or ‘national identity management systems’, even though those definitions are not identical) are receiving a mixed acceptance, with their privacy, security and usability being criticised, specifically in the UK. This paper investigates whether it is possible to improve social acceptance of such systems in cases where they are incompatible with the perceived value of privacy, but without significantly changing their original architecture. The paper analyses requirements using four different scenarios that address (...) long-term privacy issues. Relatively small alterations to such systems are suggested that may significantly improve their adoption. (shrink)

The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum (...) with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others. (shrink)

As shown in (Łukowski, Gensler, 2013), the paradox of the stone is a failed attempt to show that “omnipotence” is a contradictory concept. An element of the argument presented there is that God, while unable to lift the stone, can nevertheless annihilate it. This work considers the amplification of the paradox of the stone to the form generated by the question: can God create a stone which He will not be able to lift, nor, once created, will He be able (...) to destroy. (shrink)

The article presents a thought experiment aimed at indicating a possibility for thinking education beyond the logic of progress. In its first part, the argument reconstructs the entanglement of the modern idea of progress (as found in Francis Bacon and Comenius) and education, while tracking down the specific coupling of obedience and conquest at work. Through such an analysis a link between the ideas of progress and of emancipation is determined, which leads to the acknowledgement of the difficulty of the (...) task of imagining education outside of the logic of progress. In the second part an attempt is made to match this task by suggesting that when studying with a teacher, the logic of progress is deactivated. A phenomenological analysis of the practices involved in studying with a teacher points to a specific way of living together that such a collective study involves. This way of living together is formed by continual exercises in the humble equality realized through shared attention to something worthy of study: this stems from attentiveness to what exists, and it develops an attitude of care and respect for being as such. (shrink)

This paper discusses differences between two major schools in philosophy of criminal law, retributivism and consequentialism, with regard to the risk of punishing the innocent. As it is argued, the main point of departure between these two camps in this respect lies in their attitude towards the high evidentiary threshold in a criminal trial: while retributivism seems to strongly support setting this standard high, consequentialists may find it desirable to relax it in some cases. This discussion is set in the (...) context of proxy criminalization, i.e. a situation, in which some suspicious behaviour is criminalized. Since proxy criminalization may be understood as an effective lowering of the evidentiary threshold, its employment is justifiable from the consequentialist perspective, while being highly problematic for the retributivists. (shrink)

This theoretical paper is offered in the spirit of advancing the debate on the socioemotional wealth construct and its impact on how family firms conceptualize and practise corporate social responsibility. The study builds on Kellermanns et al.’s :1175–1182, 2012) claim that the SEW dimensions can be positively and negatively valenced as well as makes a distinction between the selective and instrumental approach to CSR and the holistic and normative one. Drawing on these considerations, it provides a theoretical underpinning in favour (...) of the view that SEW has ambivalent nature and therefore can produce detrimental outcomes for stakeholders of family companies. In this way, the study challenges the implicit assumption prevalent in the literature that SEW is “a prosocial and positive stimulus”. Crucially, it expands on the SEW construct by arguing that, given its ambivalent nature, SEW, as such, is at odds with the “strategic, whole-business view of responsibility”. Consequently, it posits that family firms—because of their concern with SEW—may be more likely to adopt the instrumental and selective rather than strategic and normative approach. Hence, it also makes the case for regarding the latter as a reference point to investigate the family company’s attitude towards social responsibility. It concludes by summarising the argument and offering future research avenues. (shrink)

The paper presents an analysis of the anumāna chapter of Jayarāśi’s Tattvôpaplava-siṁha and the nature of his criticism levelled against the anumāna model. The results of the analysis force us to revise our understanding of Jayarāśi Bhaṭṭa as a sceptic. Instead, he emerges as a highly critical philosopher. In addition, the nature of Jayarāśi’s criticism of the anumāna model allow us to conclude that anumāna should not be equated with inference, but rather is its limited subset, and may at best (...) be rendered as ‘disputational inference’, ‘debational inference’ or even ‘dialogical inference’. Jayarāśi applies a range of logical laws which clearly represent patterns of what can be classified as a priori reasoning and analytical justifications for knowledge, which were traditionally not reckoned sound. Against the backdrop of Jayarāśi’s criticism of anumāna, the paper also attempts to provide an explanation to why Indian philosophy and logic did not develop any concept of proper symbols and variables. (shrink)

The paper examines the Tattvôpaplava-siṁha of Jayarāśi Bhaṭṭa, and presents an analysis of his positive arguments that can be traced in the work. Despite the widely held opinion that Jayarāśi was a sceptic or held no positive opinions, the author concludes that, first, Jayarāśi does not fit a standard description of a sceptic. What may appear as an approach to philosophical problems, typical of a sceptic, turns out to be Jayarāśi’s particular method of critical examination on the part of a (...) rationalist. Second, a number of positive views Jayarāśi entertains can be identified in his work, and most of them overlap with much of the doctrine of the Cārvākas and Lokāyatas and materialist tradition recorded as early as the Sāmañña-phala-sutta. Therefore, Jayarāśi should be classified as a representative of the Cārvāka/Lokāyata tradition. (shrink)

Foundations of Science recently published a rebuttal to a portion of our essay it published 2 years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and intuition. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses text with context and thereby misunderstands the significance of Cauchy’s use of infinitesimals. Here we defend our original analysis of various misconceptions and misinterpretations concerning (...) the history of infinitesimals and, in particular, the role of infinitesimals in Cauchy’s mathematics. We show that Schubring misinterprets Proclus, Leibniz, and Klein on non-Archimedean issues, ignores the Jesuit context of Moigno’s flawed critique of infinitesimals, and misrepresents, to the point of caricature, the pioneering Cauchy scholarship of D. Laugwitz. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...) solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)